Phys. Fluids 8 (1996) 1350-1352.
The three-dimensional development of the shearing instability of
Department of Theoretical Mechanics, University of Nottingham,
University Park, Nottingham NG7 2RD, UK
Department of Applied Mathematics and Theoretical Physics,
University of Cambridge, Cambridge, CB3 9EW, UK
Two-dimensional convection can become unstable to a mean shear flow. In three
dimensions, with periodic boundary conditions in the two horizontal directions,
this instability can cause the alignment of convection rolls to alternate
between the x and y axes. Rolls with their axes in the
y-direction become unstable to a shear flow in the
x-direction that tilts and suppresses the rolls, but this flow does
not affect rolls whose axes are aligned with it. New rolls, orthogonal to the
original rolls, can grow, until they in turn become unstable to the shear flow
instability. This behaviour is illustrated both through numerical simulations
and through low-order models, and the sequence of local and global bifurcations
gzipped PostScript version of this paper (0.2MB)
Movies relevant to this paper: behaviour as a function of r
Numerical simulations of 3D convection in a compressible layer, in a cubic
box with periodic boundary conditions in the two horizontal directions. Fixed
temperature and stress-free boundary conditions are imposed at the top and
bottom of the box. The Prandtl number is 0.8; the controlling parameter is
Movies courtesy of Derek
- r=1.00: convection sets in as two-dimensional rolls, which
continue to r=1.20.
- By r=1.40, rolls lose stability to tilted rolls.
- These in turn lose stability to tilted squares by r=1.50.
- By r=1.55, time dependence begins in a Hopf bifurcation
leading to oscillatory tilted squares
(gzipped movie, 0.4MB).
- At r=1.65, chaos has set in near a heteroclinic bifurcation
(gzipped movie, 1.5MB).
- At r=1.80, there is a structurally stable heteroclinic cycle
(gzipped movie, 1.3MB).
- At r=2.00, the structurally stable heteroclinic cycle
connects fixed points and periodic orbits
(gzipped movie, 1.7MB).